| L(s) = 1 | − 2.18·2-s + 2.79·4-s − 0.913·5-s − 7-s − 1.73·8-s + 1.99·10-s − 2.64·11-s + 4·13-s + 2.18·14-s − 1.79·16-s − 3.46·17-s + 5.58·19-s − 2.55·20-s + 5.79·22-s + 3.46·23-s − 4.16·25-s − 8.75·26-s − 2.79·28-s + 8.75·29-s − 9.16·31-s + 7.38·32-s + 7.58·34-s + 0.913·35-s + 3·37-s − 12.2·38-s + 1.58·40-s + 0.913·41-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.39·4-s − 0.408·5-s − 0.377·7-s − 0.612·8-s + 0.632·10-s − 0.797·11-s + 1.10·13-s + 0.585·14-s − 0.447·16-s − 0.840·17-s + 1.28·19-s − 0.570·20-s + 1.23·22-s + 0.722·23-s − 0.833·25-s − 1.71·26-s − 0.527·28-s + 1.62·29-s − 1.64·31-s + 1.30·32-s + 1.30·34-s + 0.154·35-s + 0.493·37-s − 1.98·38-s + 0.250·40-s + 0.142·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5612262167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5612262167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 5 | \( 1 + 0.913T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 0.913T + 41T^{2} \) |
| 43 | \( 1 - 0.582T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 0.582T + 79T^{2} \) |
| 83 | \( 1 - 9.66T + 83T^{2} \) |
| 89 | \( 1 - 1.82T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.70801114956749144954057205965, −9.719397940962185575028927265198, −8.991035375464609158646788252235, −8.224931866690150907076087560829, −7.44078499090812850038464065049, −6.61883222423940368517297088987, −5.36320787007934773536145390377, −3.85476814935393231644238641403, −2.46418701411566740877735955625, −0.836137487424588872797300600778,
0.836137487424588872797300600778, 2.46418701411566740877735955625, 3.85476814935393231644238641403, 5.36320787007934773536145390377, 6.61883222423940368517297088987, 7.44078499090812850038464065049, 8.224931866690150907076087560829, 8.991035375464609158646788252235, 9.719397940962185575028927265198, 10.70801114956749144954057205965
